In this study, the authors verify fixed-point results for Geraghty contractions with a restricted co-domain of the auxiliary function in the context of generalized metric structure, namely the $S_b$-metric space. This new idea of defining Geraghty contraction for self-operators generalizes a large number of previously published, closely related works on the presence and uniqueness of a fixed point in $S_b$-metric space. Also, the outcomes are achieved by removing the continuity constraint of self-operators. We also provide examples to elaborate on the obtained results and an application to the integral equation to illustrate the significance in the literature.

In this paper, we extend the definition of coupled fixed point to mappings on cone \(A_{b}\)-metric space and prove some coupled fixed point theorems. Our results extend the coupled fixed point results of Singh and Singh [K. A. Singh, M. R. Singh, J. Math. Comput. Sci., \(\bf 10\) (2020), 891--905] to cone \(A_{b}\)-metric space. An example is also given to illustrate the validity of our result.

In this paper, we introduce the concept of F-contractions in parametric S-metric space. Further, we prove a unique fixed point theorem in parametric S-metric space. Another result is also obtained as corollary.